Approximations to Robust Conic Optimization Problems

2012 ◽  
Keyword(s):  
Optimization Problems ◽  
Conic Optimization

scholarly journals Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints

2021 ◽  
Author(s):  
Dimitris Bertsimas ◽  
Ryan Cory-Wright ◽  
Jean Pauphilet
Keyword(s):  
Optimization Problems ◽  
Decomposition Methods ◽  
Low Rank ◽  
Mixed Integer ◽  
Conic Optimization ◽  
Integer Optimization ◽  
New Paradigm ◽  
Mixed Integer Optimization ◽  
Rank Matrix ◽  
Rank Constraints

Many central problems throughout optimization, machine learning, and statistics are equivalent to optimizing a low-rank matrix over a convex set. However, although rank constraints offer unparalleled modeling flexibility, no generic code currently solves these problems to certifiable optimality at even moderate sizes. Instead, low-rank optimization problems are solved via convex relaxations or heuristics that do not enjoy optimality guarantees. In “Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints,” Bertsimas, Cory-Wright, and Pauphilet propose a new approach for modeling and optimizing over rank constraints. They generalize mixed-integer optimization by replacing binary variables z that satisfy z2 =z with orthogonal projection matrices Y that satisfy Y2 = Y. This approach offers the following contributions: First, it supplies certificates of (near) optimality for low-rank problems. Second, it demonstrates that some of the best ideas in mixed-integer optimization, such as decomposition methods, cutting planes, relaxations, and random rounding schemes, admit straightforward extensions to mixed-projection optimization.

scholarly journals Refining the partition for multifold conic optimization problems

Optimization ◽  
2020 ◽  
Vol 69 (11) ◽  
pp. 2489-2507
Author(s):  
Héctor Ramírez C. ◽  
Vera Roshchina
Keyword(s):  
Optimization Problems ◽  
Conic Optimization

scholarly journals Alfonso: Matlab Package for Nonsymmetric Conic Optimization

2021 ◽  
Author(s):  
Dávid Papp ◽  
Sercan Yıldız
Keyword(s):  
Barrier Function ◽  
Optimization Problems ◽  
Symmetric Cone ◽  
Conic Optimization ◽  
Iteration Complexity ◽  
Worst Case ◽  
Optimization Software ◽  
Second Order Cone ◽  
Matlab Package ◽  
Hyperbolicity Cones

We present alfonso, an open-source Matlab package for solving conic optimization problems over nonsymmetric convex cones. The implementation is based on the authors’ corrected analysis of a method of Skajaa and Ye. It enables optimization over any convex cone as long as a logarithmically homogeneous self-concordant barrier is available for the cone or its dual. This includes many nonsymmetric cones, for example, hyperbolicity cones and their duals (such as sum-of-squares cones), semidefinite and second-order cone representable cones, power cones, and the exponential cone. Besides enabling the solution of problems that cannot be cast as optimization problems over a symmetric cone, algorithms for nonsymmetric conic optimization also offer performance advantages for problems whose symmetric cone programming representation requires a large number of auxiliary variables or has a special structure that can be exploited in the barrier computation. The worst-case iteration complexity of alfonso is the best known for nonsymmetric cone optimization: [Formula: see text] iterations to reach an ε-optimal solution, where ν is the barrier parameter of the barrier function used in the optimization. Alfonso can be interfaced with a Matlab function (supplied by the user) that computes the Hessian of a barrier function for the cone. A simplified interface is also available to optimize over the direct product of cones for which a barrier function has already been built into the software. This interface can be easily extended to include new cones. Both interfaces are illustrated by solving linear programs. The oracle interface and the efficiency of alfonso are also demonstrated using an optimal design of experiments problem in which the tailored barrier computation greatly decreases the solution time compared with using state-of-the-art, off-the-shelf conic optimization software. Summary of Contribution: The paper describes an open-source Matlab package for optimization over nonsymmetric cones. A particularly important feature of this software is that, unlike other conic optimization software, it enables optimization over any convex cone as long as a suitable barrier function is available for the cone or its dual, not limiting the user to a small number of specific cones. Nonsymmetric cones for which such barriers are already known include, for example, hyperbolicity cones and their duals (such as sum-of-squares cones), semidefinite and second-order cone representable cones, power cones, and the exponential cone. Thus, the scope of this software is far larger than most current conic optimization software. This does not come at the price of efficiency, as the worst-case iteration complexity of our algorithm matches the iteration complexity of the most successful interior-point methods for symmetric cones. Besides enabling the solution of problems that cannot be cast as optimization problems over a symmetric cone, our software can also offer performance advantages for problems whose symmetric cone programming representation requires a large number of auxiliary variables or has a special structure that can be exploited in the barrier computation. This is also demonstrated in this paper via an example in which our code significantly outperforms Mosek 9 and SCS 2.

scholarly journals CQ-free optimality conditions and strong dual formulations for a special conic optimization problem

2020 ◽  
Vol 8 (3) ◽  
pp. 668-683
Author(s):  
Olga Kostyukova ◽  
Tatiana V. Tchemisova
Keyword(s):  
Optimality Conditions ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Constraint Qualifications ◽  
Strong Duality ◽  
Conic Optimization ◽  
First Order ◽  
Primal And Dual ◽  
Linear Cost ◽  
Strong Dual

In this paper, we consider a special class of conic optimization problems, consisting of set-semidefinite(or K-semidefinite) programming problems, where the set K is a polyhedral convex cone. For these problems, we introduce the concept of immobile indices and study the properties of the set of normalized immobile indices and the feasible set. This study provides the main result of the paper, which is to formulate and prove the new first-order optimality conditions in the form of a criterion. The optimality conditions are explicit and do not use any constraint qualifications. For the case of a linear cost function, we reformulate the K-semidefinite problem in a regularized form and construct its dual. We show that the pair of the primal and dual regularized problems satisfies the strong duality relation which means that the duality gap is vanishing.

scholarly journals Solving Conic Optimization Problems via Self-Dual Embedding and Facial Reduction: A Unified Approach

2017 ◽  
Vol 27 (3) ◽  
pp. 1257-1282 ◽  
Author(s):  
Frank Permenter ◽  
Henrik A. Friberg ◽  
Erling D. Andersen
Keyword(s):  
Optimization Problems ◽  
Conic Optimization ◽  
Unified Approach ◽  
Facial Reduction
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